{
  "id": "math/0511054",
  "version": "v1",
  "published": "2005-11-02T16:57:12.000Z",
  "updated": "2005-11-02T16:57:12.000Z",
  "title": "The Averaging lemma and regularizing effect",
  "authors": [
    "Eitan Tadmor",
    "Terence Tao"
  ],
  "comment": "28 pages; no figures; submitted, Comm. Pure. Appl. Math",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove new velocity averaging results for second-order multidimensional equations of the general form, $\\op(\\nabla_x,v)f(x,v)=g(x,v)$ where $\\op(\\nabla_x,v):=\\bba(v)\\cdot\\nabla_x-\\nabla_x^\\top\\cdot\\bbb(v)\\nabla_x$. These results quantify the Sobolev regularity of the averages, $\\int_vf(x,v)\\phi(v)dv$, in terms of the non-degeneracy of the set $\\{v: |\\op(\\ixi,v)|\\leq \\delta\\}$ and the mere integrability of the data, $(f,g)\\in (L^p_{x,v},L^q_{x,v})$. Velocity averaging is then used to study the \\emph{regularizing effect} in quasilinear second-order equations, $\\op(\\nabla_x,\\rho)\\rho=S(\\rho)$ using their underlying kinetic formulations, $\\op(\\nabla_x,v)\\chi_\\rho=g_{{}_S}$. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, non-isotropic diffusion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-02T16:57:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L65",
      "35K57"
    ],
    "keywords": [
      "regularizing effect",
      "averaging lemma",
      "nonlinear conservation laws",
      "quasilinear second-order equations",
      "elliptic equations driven"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11054T"
    }
  }
}